Altered Form is the classic way to do this as reference in Thaumatology under Spirit Vessel and also in Madness Dosier.
Basically the GM designs or approves a racial template representing the spirit possession.

This seems to be what I'm envisioning. The possessing spirit is already coming with a were form so don't really want to add another one on top of it just for the personality.

Why not? If you want a different set of skills for the spirit, this may be the only way about it, actually. On the plus side, this form would be discounted because there's no disguise element, unlike a normal Alternate Form.

I'm trying to figure out how to properly stat out a character who thanks to exposure to a magical relic now periodically becomes possessed by the spirit/avatar of a god.
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Which does what? Does he have any control of or memory of what it does? Does the player get to to control the avatar? Is it triggered by stress or by the god's own agenda when it sees an opportunity to put a spoke in another god's wheel or help its own cult?

Which does what? Does he have any control of or memory of what it does? Does the player get to to control the avatar? Is it triggered by stress or by the god's own agenda when it sees an opportunity to put a spoke in another god's wheel or help its own cult?

The avatar would have a general sense of the 'host's' memories.

Full bio: The player should eventually encounter a magical artifact that will grant them the ability to shapeshift into a new form. While I want them to have access to the form on their own, it comes with a drawback. The driving will of the avatar spirit trying to exercise it's own influence on the body. When it does, the player will be unaware of what the spirit does, but is aware of its existence. The spirit will have control of the players abilities plus a fraction of its own, with its own greater agenda separate from the players.